What Are the Odds?

I get a lot of e-mail. I eventually respond to everyone, but I do an extremely poor job of responding in a timely fashion. So, I thought I'd do a column that generally answers a question I often receive in various forms. The question goes something like this: "I had such and such a hand, and it was a really good hand, but my opponent beat me by having/making a better one. What are the odds of this?"



At this point, you may be wondering how I'm going to answer the question, "What are the odds?" for many situations all in this space. Well, I'm not. In fact, the idea I'd like to get across here is that "what are the odds?" is not a useful question for most poker discussions, and in fact it's often an unanswerable question. Let me give you an example.



A friend of mine recently asked me about a hand in which a player had an ace-high flush, but his opponent beat him with the K Q for a straight flush. Our hero had the A 8, and the board contained the J 10 9. It was an unpaired board, so the K Q was literally the only hand that beat him.



"What were the odds of that guy having the king-queen of diamonds?" my friend asked.



I cringed. "Well, what do you mean?" I asked.



"You know; what were the chances that this particular player had been dealt the king-queen of diamonds?"



"OK," I said, and then after some thinking, contiuned, "that's 1-in-1,225, given the two cards we have." I then tried to explain that even though the chances of this particular opponent holding the K Q were 1-in-1,225, any player at the table who'd been dealt the K Q would've played it, which changed the odds significantly. I also mentioned that all of the boardcards had to be considered.



Unfortunately, it later got back to me that I had listened to the details of the hand, and advised calling with the ace-high flush because I had (supposedly) said that the chances of the opponent having the K Q were 1-in-1,225. In fact, even without knowing any of the action, I knew the chances that the enemy had the K Q were much better than 1-in-1,225. To show this, let's try to make an estimate of what the actual odds were.



First, we'll do some calculating without considering any of the action. It was a sixhanded game, so that means the chances that one of the five opponents was dealt the K Q were about 1-in-250, or 0.4 percent. (This is roughly five times the chance that one specific opponent was dealt the K Q.) But given that the hand got to the river with no K or Q on board, we can actually say that there were only 45 unknown cards, making the chances of any specific player having been dealt the K Q only 1-in-990, and the chance that someone had been dealt the K Q about 1-in-200. So, clearly, the 1-in-1,225 number isn't close.



But all of this mathematics completely misses the point. A poker mathematician (and I am far more comfortable calling myself a "poker mathematician" than a real mathematician) would consider the action first and work backward. So, what was the action? I later learned that the opponent in question called a $2 raise preflop, checked the J 10 9 flop (our hero checked, as well), checked the blank on the turn (our hero also checked), bet $5 on the river, and then moved in for $75 after our hero raised to $15.



It might be slightly unorthodox to analyze the hand in this fashion, but let's begin at the river. With what hands would typical players bet the river, and then reraise all in on that board? Certainly, the K Q is one such hand. An ace-high flush and the 8 7 are some other hands, but those are impossible, because our hero had the A 8. Many players would also make this reraise with a king-high flush. Some players would even make the play with a smaller flush or a straight, but at this point, I should tell you that my friend described his opponent as "tight," and thus ruled out hands such as those. But now let's go back to the earlier action. If this player was tight, there are very few suited kings he could have. The K J, K 10, and K 9 are all impossible because of the boardcards. The K 8 is impossible because our hero has the 8, and the K 7 or worse is not a hand with which tight players typically call preflop raises. So, really, the only hand I would expect this tight player to reraise for value is, in fact, the K Q.



So, the next question is, would this tight player ever bluff? Well, without further information (and by the way, every opinion I have about every poker situation is tacitly preceded by the clause "without further information"), I'd say this player would occasionally bluff, because nearly everyone bluffs sometimes. They wouldn't be poker players if they didn't. But the most logical hand for him to bluff with is the naked A, and again, that's impossible because of our hero's hand. What, then, is the percent chance that this player has decided to try a wild bluff with something like 7-7 or 5-5, or possibly the naked Q or K? That question, and not a many-decimal-point calculation of how likely our opponent was to have been dealt the K Q before anything else happened, is the key to deciding whether to call – because it does really seem unlikely that our ace-high flush beats any value-reraises here.



It's funny, in doing my initial analysis, I told my friend that the A 8 should call with his second-nut hand. After looking at it more closely (and, of course, learning what the action was, and getting some information about the player), I put the chances that this tight opponent in a low-limit game had precisely the K Q at about 80 percent, which makes it a clear fold for our hero's ace-high flush. But believe it or not, learning to make big folds is not going to be the difference in turning an average poker player into a good poker player, or even a good poker player into a great one. Rather than trying to figure out when he should fold the second nuts, the player with the A 8 would be better served by asking himself why he decided to slow-play all the way to the river, and why he thinks his slow-play will win him the most money in the long run, but that is a subject for another column.

Matt Matros is the author of The Making of a Poker Player, which is available online at http://www.cardplayer.com/.